1. Field of the Invention
This invention relates to improved Wavefront Coding Optics for controlling focus related aberrations, and methods for designing such Wavefront Coding Optics.
2. Description of the Prior Art
Wavefront Coding is a relatively new technique that is used to reduce the effects of misfocus in sampled imaging systems through the use of Wavefront Coding Optics which operate by applying aspheric phase variations to wavefronts of light from the object being imaged. Image processing of the resulting images is required in order to remove the spatial effects of the Wavefront Coding. The processed images are sharp and clear, as well as being relatively insensitive to the distance between the object and the detector.
Wavefront Coding is also used to control general focus related aberrations to enable simplified design of imaging systems as well as to provide anti-aliasing in sampled imaging systems.
The Wavefront Coding Optics taught and described in prior art, beginning with U.S. Pat. No. 5,748,371, issued May 5, 1998, were discovered by trial and error. The first operative Wavefront Coding mask applied a cubic phase function to the wavefront from the object. It was known that Wavefront Coding Optics, like the cubic mask, needed to apply aspheric, asymmetric phase variations to the wavefronts.
Prior art related to Wavefront Coding systems includes a fundamental description of Wavefront Coding (U.S. Pat. No. 5,748,371), description of Wavefront Coding used for anti-aliasing (Anti-aliasing apparatus and methods for optical imaging, U.S. Pat. No. 6,021,005, Feb. 1, 2000), use of Wavefront Coding in projection systems (Apparatus and methods for extending depth of field in image projection systems, U.S. Pat. No. 6,069,738, May 30, 2000), and the combination of Wavefront Coding and amplitude apodizers (Apparatus and method for reducing imaging errors in imaging systems having an extended depth of field, U.S. Pat. No. 6,097,856, Aug. 1, 2000).
The layout of a conventional Wavefront Coded imaging system is shown in FIG. 1. Imaging Optics 104 collects light reflected or transmitted from Object 102. Wavefront Coding Optics 106 modify the phase of the light before detector 108. Wavefront Coding Optics 106 comprise a cubic mask. Detector 108 can be analog film, CCD or CMOS detectors, etc. The image from detector 108 is spatially blurred because of Wavefront Coding Optics 106. Image processing 110 is used to remove the spatial blur resulting in a final image. i.e., image processing 110 removes the Wavefront Coding applied by optics 106, thereby reversing the effects of optics 106, other than the increase in depth of field and depth of focus. The image before and after Image Processing 110 also is very insensitive to misfocus aberrations. These misfocus aberrations can be due to the Object 102 being beyond the depth of field of the Imaging Optics 104, the detector 108 being beyond the depth of focus of the Imaging Optics 104, or from Imaging Optics 104 having some combination of misfocus aberrations such as spherical aberration, chromatic aberration, petzval curvature, astigmatism, temperature or pressure related misfocus.
FIG. 2 describes a rectangularly separable prior art Wavefront Coding phase function that produces an extended depth of field. This phase function is a simple cubic phase function that is mathematically described, in normalized coordinates, as:cubic-phase(x,y)=12 [x3+y3]|x|≦1, |y|≦1
Other related forms of the cubic mask are described as:cubic-related-forms(x,y)=a[sign(x)|x|b+sign(y)|y|b],where|x|≦1, |y|≦1,andsign(x)=+1 for x≧0, sign(x)=−1 otherwise
These related forms trace out “cubic like” profiles of increasing slopes near the end of the aperture.
The top plot of FIG. 2 describes a 1D slice along an orthogonal axis of the cubic phase function. The lower plot of FIG. 2 describes the contours of constant phase of this cubic phase function.
FIG. 3 shows MTFs as a function of misfocus for a system with no Wavefront Coding and for a system with the conventional Wavefront Coding cubic phase function of FIG. 2. The normalized misfocus values are the same for both systems and are given as ψ={0, 2, 4}, where ψ=[2 pi W20], and where W20 is the conventional misfocus aberration coefficient in waves. MTFs with no Wavefront Coding (302) are seen to have a large change with misfocus. MTFs with the rectangularly separable cubic phase function (304) are seen to change much less with misfocus then the system with no Wavefront Coding.
A non-separable prior art form of Wavefront Coding Optics, in normalized coordinates, is:non-separable-cubic-phase(p, q)=p3 cos(3 q) |p|≦1, 0≦q≦2pi
This phase function has been shown to be useful for controlling misfocus and for minimizing optical power in high spatial frequencies, or antialiasing. When using a digital detector such as a CCD or CMOS device to capture image 108, optical power that is beyond the spatial frequency limit of the detector masquerades or “aliases” as low spatial frequency power. For example, say that the normalized spatial frequency limit of a digital detector is 0.5. As seen from FIG. 3, the in-focus MTF from the conventional system with no Wavefront Coding can produce a considerable amount of optical power beyond this spatial frequency limit that can be aliased. By adding misfocus to the system without Wavefront Coding the amount of high spatial frequency optical power can be decreased, and aliasing reduced, as is well known. When using conventional Wavefront Coding, as shown in FIG. 3, the amount of optical power that can be aliased can be decreased (304) compared to the system without Wavefront Coding (302).
Image Processing function 110 essentially applies amplification and phase correction as a function of spatial frequency to restore the MTFs before processing to the in-focus MTF from the conventional system with no Wavefront Coding after processing, or to some other application specific MTF, if desired. In effect, the Image Processing function of FIG. 1 removes the Wavefront Coding blur in the detected image.
In practice the amplification applied by the Image Processing function increases the power of the deterministic image but also increases the power of the additive random noise as well. If Image Processing 110 is implemented as a linear digital filter then a useful measure of the increase of power of the additive random noise is called the Noise Gain of the digital filter. The concept of “noise gain” is commonly used in radar systems to describe the amount of noise power at the output of radar digital processors. Nonlinear implementations of Image Processing 110 have similar types of noise-related measures. The Noise Gain for a digital filter is defined as the ratio of the root-mean-square (RMS) value of the noise after filtering to the RMS value of the noise before filtering. In general the Noise Gain is nearly always greater than one in Wavefront Coded systems. Assuming that the additive noise is uncorrelated white gaussian noise, the Noise Gain of a two dimensional linear digital filter can be shown to be equal to:Noise Gain=sqrt[ΣΣf(i,k)2]=sqrt[ΣΣ|F(wi,wk)|2]where[ΣΣf(i,k)]=F(0,0)=1.0,f(i,k) is a spatial domain digital filter, F(wi,wk) is the equivalent frequency domain digital filter, and the first sum is over the index i or k and the second sum is over the other index. Indices (i,k) denote spatial domain coordinates while indices (wi,wk) denote frequency domain coordinates. The constraints that the sum of all values of the filter and the zero spatial frequency filter value both equal unity ensures that the zero spatial frequency components of the image (the background for example) are unchanged by the image processing.
Wavefront Coded MTFs that have the highest values require the least amplification by the digital filter and hence the smallest Noise Gain. In practice the Wavefront Coding Optics that produce MTFs that have small changes over a desired amount of misfocus and also have the highest MTFs are considered the best and the most practical Optics for Wavefront Coding. Optics that produce MTFs that have small changes with misfocus but also very low MTFs are impractical due to very large Noise Gain of the resulting digital filters. Digital filters with large Noise Gain will produce final images that have unnecessarily high levels of noise.
While the conventional cubic Wavefront Coding mask does operate to increase depth of field and control focus related aberrations, there remains a need in the art for improved Wavefront Codings Optics, which retain the capacity to reduce focus-related aberrations, while also producing high value MTFs. There also remains a need in the art for methods of designing such improved Wavefront Coding Optics.